Hi I'm Melody with Magoosh and today I want to share some tips that can help you prepare for
and address exam questions that have to do with geometry on the GMAT. Now we might have
taken geometry class a very long time ago, I know I did, and so if those ideas haven't stayed fresh
in your mind and you haven't used them, there could be a little amount of review that you need to do.
I'm happy to tell you, you don't need to learn an entire year's worth of geometry class all
over again, but there are some basic principles and some fundamental things that you do need to
know so you can apply the right properties and formulas on the exam. So we do need to do some
reviewing, some remembering. Let's talk about the most common shapes tested on the exam. And as we learn
about those shapes we want to pay attention primarily to angle measurements, relationship between angles,
relationships between angles and sides of the figure,
we always want to be able to find the area of a figure for a two-dimensional figure or the volume for a 3D one
and the perimeter. Now perimeter is my favorite, it's the easiest because if all the sides
of the shape are straight lines, which most of our shapes are, then we just add those sides up and we
have the perimeter. So let's look at what that might look like as we dig into the details of
one of the most familiar shapes that we see in the world all around us, the circle. Now the circle
is going to be a little bit different from the others because it doesn't have straight line sides.
So let's look at how we can analyze the most important facts about circles for the exam.
Circle is defined as the set of all points that are the same distance from the center point.
So circles are usually named with their center very often not always but very often that center
point will be called "o" just because that's a common practice. Every point on the outside has
the same distance from that center and that distance is called the radius. If we take two
radii, one radius and then another straight across, then that length is called the diameter. Here's how
those are spelled: radius, diameter. And the basic most fundamental circle formulas that we use are
based right off of that relationship, the radius and the diameter. So for a circle the area equals
pi times the radius squared. The circumference is a special word we use in circles for the distance
around the outside it's very similar to the perimeter but technically perimeter isn't the most
accurate term because there are no straight line sides here but the distance all the way around the
outside of the circle if you walked completely around the edge that's called the circumference.
And we have two formulas for circumference that mean the same thing one is 2 pi times the radius
the other is pi times the diameter. Notice that makes sense because two times the radius is the
same as the diameter we saw that in our defining diameter and in fact that's another formula
that we could remember and be ready to use on the exam. There are some other pieces of circles
that can be tested for example if we take two radii we cut a piece of that circle that's a wedge
almost like a slice cut out of a pizza on the inside of that piece there's an angle right there
that's called the central angle of the circle. The total number of degrees in a circle is 360
degrees so that angle is going to be a portion of those 360 degrees. For example let's say that our
angle here is 60 degrees, that means that that slice or that wedge, which is called a sector,
represents 60 degrees out of 360 total degrees or exactly 1/6 of the circle,
in our case since we have a 60 degree angle. That gives us a couple of extra pieces of information.
For example let's say we wanted to travel along the circumference
of this circle from one end of that sector to the other. This is a piece called the arc
of the circle, an arc. And it has a measurement called arc length. The arc length
is equal to the total circumference of the circle times the degree measurement of that central angle
divided by our total 360 degrees. In this case if we knew the circumference of the circle
we'd multiply it by 1/6 and we'd have the arc length. We could also be asked to find the area
inside that sector. So the area of a sector sometimes we use the phrase "sector area"
is equal to the total area of the circle times again the central angle
over 360 degrees. In this case we have a 60 degree central angle so we could take the total area of
that circle multiply it by 1/6 and we would find the area of the sector. Okay so notice we didn't
learn every detail in the world about circles but we have a good overview of all of the primary
properties about them and that's the same way you want to approach your review of other shapes as
well. Personally for other shapes I think it's easy to organize them by the number of sides in
the shape and we're going to talk about straight line sides and two dimensional shapes. For example
one line/one side is just a straight line and there actually are some properties regarding
angles when you see a straight line. Two lines when they cross make an angle as well and so you want
to pay attention to when two lines intersect each other what kind of angle properties and
relationships there are. Three lines together makes a triangle and that's one of the most
important shapes you can study for the exam. Standardized tests love triangles because there
are some special properties to them. Especially you want to pay attention to what's called "special
right triangles". So a right triangle is one that has a right angle a 90 degree angle somewhere in
that triangle and there are some properties of certain types of right triangles that are often
tested on the GMAT. So you want to pay attention to 30, 60, 90 degree right triangles 45, 45, 90 degree
right triangles and then anything that satisfies the pythagorean theorem in a special way is likely
to show up on the exam. Pythagorean theorem has a relationship between the three sides of a triangle
remember, a squared plus b squared equals c squared and for right triangles only. Now if those three a
b and c all happen to be whole numbers integers that's something pretty rare and pretty special.
And so when that happens it's called a pythagorean triple for example 3, 4, 5 is the
most common pythagorean triple that you'll see on the exam so look out for those on the test as well.
When we have four sides connected that makes it quadrilateral. The common ones we use in everyday
language might be a rectangle or a square. We also want to learn about parallelograms,
trapezoids, different relationships between the angles the sides in those shapes.
Once we go past four sides we can keep adding a number of sides forever but there aren't so
many special properties that will be tested on the exam. But for five or more sides you do
want to know what the word "regular" means when it refers to a shape or a polygon when one is regular
or "not regular" and you want to be able to find information about the angles inside that shape.
So that's your checklist of things to learn and review once you've done that it's time to actually
work on test questions. So as you're practicing test questions and building up a strategy for
test day here are some tips that I recommend. First, you always want to work from an image on
your scratch paper. So it's possible when you see a geometry question there's a figure on the screen
if so copy it down sometimes the geometry is just being described in words though so
you want to make the best sketch that you can of what you're reading about. It does not have to be
perfect circles do not have to be perfectly round lines do not have to be perfectly straight but it
should reasonably reflect what's going on. Remember it's your scratch paper nobody's going to check it,
nobody's going to grade you on it, but it should it should help you work toward the correct answer.
Then when you've got that drawn you want to label all of the side lengths and angle measurements
and information that you're given right onto that diagram. So that you're working off of
that image on your scratch paper. Geometry is a very visual part of mathematics so you want to
let that work to your advantage. One word of warning here you should not assume the diagrams
you see on the screen are exactly the right shape and size. For example it might look like two side
lengths are the same but one might actually be bigger than the other in real life or it might
look like one angle is way bigger than another but it turns out they're the same so that's a caution.
Diagrams might not always be drawn to scale is what it's called so it might not be exactly the
right relationship, don't rely on eyeballing or estimating but instead rely on actual numbers
that you're given to work out the problem. I also recommend writing down any formulas that you can
tell you're going to need to solve the problem on your scratch paper. That can also help you.
have everything you need organized right there in front of you. For example if you know you're going
to need to use the formula for area of a triangle write it down on your scratch paper and that's one
less thing to worry about, have everything right in front of you that you can work from as a resource.
Now let's say you've got all of that but you don't see the path to the correct answer. It
might be asking for some information that you just don't see built into what's given in the question.
It's best not to panic don't freeze don't get stuck sitting there doing nothing.
Instead look at the information you have that's given to you and say
"can I figure some detail out based on this start?" figuring out the pieces of information you can
and each of those will be like a stepping stone putting you on the path toward the correct answer.
Let's try to put all this into practice on an example problem together going back to our old
friend the circle. Here we have a multiple choice geometry problem. Let's take a look,
each circle has center o so that word "each" means every one that means there's more than one. We'll
look out for that. Let's look at our image, well do I see two different circles? Yeah I see the one on
the inside with a white area filled in and then I see a bigger circle formed by the shaded area.
So what I would do first on my scratch paper is copy this down the best I can there is no
way my circle is going to be perfect but it's my scratch paper and nobody's going to judge me for
that. So I'm just going to do my best and I will kind of shade in this outer area, I'm not going
to take time to color it in very carefully. So each circle has center o and we see that labeled.
The radius of the smaller circle is 2. Okay so I'm going to mark that from here to here
is 2. Notice I'm taking the information from the written problem transferring it onto my drawing,
on my scratch paper, and the radius of the larger circle is 6. Here's where it works out well that
I did not fill in that circle all the way. It makes it easier for me to mark that that is
six. Here's the question, if a point is selected at random so I'm starting to get this feeling of
probability going on over here so look out for that. If a point is selected at random from the
larger circular region, so the whole big thing. What is the probability we were right that the
point will lie in the shaded region? Okay so quick thought about probability, probability
equals the number of desired outcomes over the total number of possible outcomes. In this case
it's going to be the probability of landing in the shaded region divided by the probability of the
large circular region, we'll call it large circle. Okay so that's what I need to find. Let's think
about that well I want to find a point in that location we're looking at areas inside of the
circle and I just gave the answer away we're really looking at area of a circle here. So I'm
going to write down also the formula area equals pi r squared so that I don't have to second guess,
I don't have to remember it's there ready for me to use on my scratch paper. So what are we going to
do? We need to find the area of the big circle, we need to find the area of the shaded region,
that might seem complicated. We don't really have an area formula for sort of a ring around
a circle but not a full circle. Here's a tip that can un-complicate everything here. Whenever we
have a shaded region as part of a geometric figure we can say that the total area equals the shaded
area plus the unshaded area. So let's work with that my total area
is the area of the whole big circle and that big old circle has a radius of six, remember? So
that area is going to be pi times six squared we don't know the area of the shaded circle yet.
That's what we're going to find using our little formula here, the unshaded area is the
inner circle and we saw that the smaller circle has a radius of 2. So its area is going to be pi
times 2 squared and we do just the tiniest bit of multiplying here, six squared gives us 36 pi.
That equals the shaded region plus four pi and so that means that our shaded area
equals 32 pi. This is a good time just to notice that the shaded area is way, way, way, way bigger
than the unshaded area, and that's not really what it looked like in the original figure was it?
So this is a good reminder for us don't trust what sizes look like in a figure drawn on the exam. Okay
now we're ready to find our total probability. So our probability of landing in the shaded region
is going to be the area of the shaded region 32 pi over the total area of the bigger circle.
We actually found that as an intermediate step didn't we? The total area we found was 36 pi
so we want to simplify this fraction the pi's cancel out both of these can be divided by
the common factor of four and we're going to get the final answer, the probability is eight ninths
answer choice e. And there we have it, we've got another question correct
and we're one question closer to an excellent score on the GMAT and to reaching our goals.
So in review, we will need to relearn some of the information from geometry class, it might have been
a few years back but we've got a basic central set of information that we need to learn. And then
we'll be set up for success once we're actually working on a problem. We want to draw an accurate-ish
diagram on our paper not rely on eyeballing or estimating sizes and shapes but rely on all
the numbers given to us as well as the formulas that help us work out details about that shape.
And if we're stuck instead of just standing there staring doing nothing it's best to start working
out the information that you can that'll put you on the path to right answers. And, I wish you
great success in your preparation and on actual exam questions based in geometry on the GMAT.